Collective Proofreading and the Optimal Voting Rule*

Duk Gyoo Kim† Jinhyuk Lee‡ Euncheol Shin§

December 29, 2019

Abstract

Policy decisions often involve a repeated proofreading process before implementation.

We present a dynamic model of proofreading decisions by a heterogeneous committee,

in which the committee decides when to stop proofreading and implement a risky pol-

icy. A proofreading process is costly but necessary because the risky project contains an

unknown number of errors, and the value of the policy decreases by the number of un-

detected errors. The proofreading process continues as long as a qualified majority votes

for continuation. Once the proofreading process ends, members receive heterogeneous

penalties based on the remaining errors. We characterize the optimal voting rule given

a joint distribution of the costs and penalties of the committee. We find that any qual-

ified voting rule for proofreading results in an inefficient outcome. Unlike the result in

Strulovici (2010), majority rule could have a bias not only toward under-experimentation

but also toward over-experimentation.

JEL Classification: D71; D72; D83

Keywords: Collective decision; Optimal proofreading; Optimal voting rule; Qualified

majority rule; Representative agent.

*We thank Hans Peter Grüner, Jinhee Jo, Charles Louis-Sidois, and Kirill Pogorelskiy for their help-ful suggestions and comments. Duk Gyoo Kim greatly acknowledges financial support from the DeutscheForschungsgemeinschaft (DFG, German Research Foundation)–Project-ID 139943784–SFB 884 at the Uni-versity of Mannheim. Jinhyuk Lee’s work is supported by a Korea University Grant (K1710031).

†Department of Economics, University of Mannheim, Germany. d.kim@uni-mannheim.de.‡Department of Economics, Korea University, Republic of Korea. jinhyuklee@korea.ac.kr.§College of Business, Korea Advanced Institute of Science and Technology, Republic of Korea.

eshin.econ@kaist.ac.kr.

1

1 Introduction

On January 28, 1986, the Space Shuttle Challenger was destroyed 73 seconds after lift-

off. The engineers warned that cold weather conditions might cause an O-ring seal to mal-

function, but the launch, which had been delayed several times, could not be delayed further.

All seven crew members were lost, and more than 4 billion dollars were wasted. We do not

know the actual decision process of the launch, but it is clear that the decision is collectively

made, and each committee member’s interests are heterogeneous. This paper concerns a

proofreading process before implementing a new policy or a project.

For implementing a new policy, agencies frequently run a proofreading process to verify

its effectiveness. A primary goal of the proofreading process is to find and fix errors as

many as possible to make sure that the policy is successful. For example, suppose that an

environmental protection agency concerns about a new manufacturing process which may

be harmful to the environment. The agency may conduct a series of investigations to detect

violations of environmental laws such as overuse of toxic chemicals and pollutants in the

process. The agency approves the manufacturing process only if the agency expects that

the costs of further investigations outweigh the benefits of detecting potentially unobserved

violations.

More complicated yet, the proofreading decisions are often made by a committee. When a

decision committee consists of heterogeneous members, some members may want to conduct

a stringent proofreading process, although it would incur a high cost of an investigation.

On the contrary, others may be less involved with the investigation and may want fast

approval of a new policy to enjoy potential benefits from policy implementation. To resolve

this conflict of interest, the committee members might impose a collective decision rule,

which is supposed to be designed to maximize social welfare. To understand the incentive

structure of such a collective proofreading process and find an optimal voting rule, we build

a model of collective proofreading decisions by heterogeneous members.

In our model, there is a committee consisting of n members, and it sequentially decides

whether to continue a costly proofreading process or to stop it and implement the policy. The

policy is risky in that it contains errors, and its value decreases by the number of undetected

errors. The committee continues the proofreading process as long as a qualified majority of

members agrees to do so.1

1It would be more realistic to consider two-stage voting procedures: one vote for the proofreading stringency,and the other one for approval of a (potentially) risky policy over a status quo policy. We mainly focus on theinefficiency involved in the proofreading stage, but we admit that additional voting stage with a status quo

2

We first consider a situation in which the errors are realized from a Poisson distribution

so that the proofreading level preferred by each member can be constant regardless of the

number of previously detected errors. To derive a clear policy implication, we assume fur-

ther that there is at least one member whose optimal proofreading level as a single decision

maker coincides with the socially optimal level. Hence, it is optimal for the committee to

delegate the decision to this representative member. However, the representative member’s

preference over the optimal number of proofreading steps can be different from the commit-

tee’s qualified number of proofreading steps. Due to this discrepancy, any qualified majority

rule may result in a socially sub-optimal outcome.

The optimal voting rule that maximizes social welfare varies by the nature of the het-

erogeneity that each committee member has. By letting e be the number of members who

prefer to stop the proofreading process no earlier than the representative member, we find

that the e-qualified majority rule is welfare-maximizing. If such e is less than half of the full

committee, then it means the proofreading procedure should continue even when a minor-

ity of members want to continue under the optimal voting rule. In order words, a majority

rule may have a bias toward under-experimentation, which is along with the main finding

of Strulovici (2010). However, we also find that a bias toward over-experimentation could

exist. In this case, a supermajority rule is optimal, and the proofreading experimentation

should be carried more conservatively. A simple majority rule could be socially optimal, but

the conditions for it are on weak ground.

We extend our model to address a situation where there is a fixed number of issues that

need to be evaluated. In particular, we assume that the errors are drawn from a Binomial

distribution. In this case, the probability of detecting additional errors depends on the num-

ber of previously detected errors, so the optimal stopping decisions are no longer stationary.

We can still characterize each committee member’s optimal stopping rule as a function of

the detected errors with respect to the proofreading levels. The same results as the Poisson

model are retained.

1.1 Related Literature

Our model builds on seminal work on the single decision maker’s optimal proofreading

problem by Yang et al. (1982). Previous studies on optimal proofreading decisions (Chow

and Schechner, 1985; Ferguson and Hardwick, 1989; Dalal and Mallows, 1988) have focused

on a single decision maker’s problem in which there are no concerns about the conflict of

policy can be another cause of potential inefficiency.

3

interest between heterogeneous committee members. In this paper, benefiting from a simple

dynamic nature under the assumption of Poisson prior distribution of errors, we extend to a

collective proofreading model without other behavioral and strategic concerns in collective

decisions such as present-bias (Jackson and Yariv, 2015) or free-riding problems (Keller and

Rady, 2010). As a result, we obtain a precise characterization of the welfare-maximizing

voting rule.

This paper is also related to the recent literature on collective decisions in dynamic set-

tings (Strulovici, 2010; Chan et al., 2018; Jackson and Yariv, 2015; Keller and Rady, 2010;

Lizzeri and Yariv, 2017). In a modeling perspective, we consider agents who do not discount

the future. This assumption is necessary for the existence of a representative agent in a dy-

namic model, as shown by Jackson and Yariv (2015). In our model, having a representative

agent is not crucial, but it renders a more precise illustration of the welfare loss of a qual-

ified majority rule compared with the socially optimal voting rule. In a setting of collective

deliberation, Lizzeri and Yariv (2017) compare the performance of different voting rules in

terms of the welfare of the committee under the presence of a self-control problem, while

we find an optimal voting rule without a self-control problem. Strulovici (2010) considers

situations where individuals learn their type more accurately by experimentation, but our

model deals with experimentation decisions under complete information about the commit-

tee’s heterogeneity. While the main finding of Strulovici (2010) is that majority rule has a

downward bias in terms of experimentation, we found that majority rule could also yield

a bias toward over-experimentation in our context. This observation is distinct from the

recent studies which attempt to link collective decisions with present-biased outcomes.

In the sense that the committees decide to stop searching for potential errors by proof-

reading, this paper is also related to studies on collective search by committees. In the

context of accepting a proposal or searching for alternative ones, Compte and Jehiel (2010)

examine how each committee member affects the set of possible agreements. We could

conduct the same exercise in our context, but our focus is more on the description of the so-

cially optimal voting rule. Albrecht et al. (2010) compare how the collective search problem

differs from a single-agent search problem with a focus on the case of symmetric agents,

whereas our paper explicitly considers heterogeneous agents. In Moldovanu and Shi (2013),

a committee decides whether to accept the current alternative with multiple attributes or to

continue the costly search, and each committee member can privately assess the quality of

only one attribute. Our model can be considered as a costly search of alternatives (with one

attribute) whose quality is nondecreasing over time. Although information aggregation and

4

adverse selection problems under private information of committee members as in Lauer-

mann and Wolinsky (2016) are worth being investigated, we focus on the case of complete

information.

We claim our paper contributes to advance theoretical models on environmental pol-

icy choice in different economic settings. Viscusi and Zeckhause (1976) provide a tractable

framework to deal with a situation in which there is no clear ranking among environmental

policies due to the presence of uncertainty. In particular, their model conveniently exam-

ines how a chance of irreversibility affects policy choice, as discussed in Viscusi (1985).

Wirl (2006) examines how irreversibilities of the environmental policy affect the optimal in-

tertemporal accumulation of greenhouse gases in the atmosphere under uncertainty. Gsot-

tbauer and van den Bergh (2011) investigate the relationship between environmental policy

decisions and other-regarding preferences. In our paper, we consider a different nature of

the policy implementation: the uncertainty of the benefit (damage) of the policy and con-

flicts of interest between heterogeneous committee members resolved by voting. Although

applications in environmental policy decisions motivate this paper, the model is applicable

in many other cases such as optimal R&D investments (Moscarini and Smith, 2001; Weeds,

2002) decided by a committee.

2 Benchmark Model

In this section, we describe a single decision maker’s decision problem. We introduce key

assumptions simplifying the problem. Especially we assume that the number of errors is

drawn from a Poisson distribution. Then, we investigate an optimal proofreading strategy,

which is a building block for analyzing the equilibrium behavior in the committee decision.

2.1 Setup

There is a single risky project containing M errors, where M is distributed over N0 with

E[M] < ∞.2 There is a single decision-maker who tries to find and correct errors through

a proofreading process. In each period t, the decision maker decides whether to stop or

continue the proofreading process. If he decides to stop the process, he pays a penalty of

D > 0 for each remaining error. Each error-finding step incurs a cost c > 0. To ignore the

time preference of the decision maker, we assume that he does not discount the future, or

2Throughout the paper, N0 represents the set of natural numbers including zero.

5

assume that the time gap between each proofreading step is minuscule.

We denote by X t the number of detected errors in period t, where X t ≥ 0 and∑t

j=1 X j ≤Mfor all t ≥ 1. We assume the following for the number of errors M and the sequence of

detected errors (X t)t≥1:

Assumption 1. For M and (X t)t≥1,

(1) the number of errors follows a Poisson distribution with λ> 0:

M∼ Poisson(λ).

(2) the number of detected errors in period t+1 follows a Binomial distribution:

Xt+1|M, X1, . . . , X t ∼ Binomial

(M−

t∑j=1

X j, p

),

where p ∈ (0,1) is the probability of detecting an error.

As shown in Lemma 3.1 in Ferguson and Hardwick (1989), Assumption 1 dramatically

simplifies the dynamic nature of the problem as the conditional probability of having errors

in period t+1 is independent of the history of previous error findings, which is recapitulated

by the following lemma.

Lemma 1. The number of remaining errors after t steps of proofreading follows a Poissondistribution with λ(1− p)t:

M−t∑

j=1X j ∼ Poisson(λ(1− p)t).

Lemma 1 implies (1) that the mathematical expectation of the number of remaining

errors is λ(1− p)t, which is decreasing in t, and (2) that the expectation is independent of a

history of error detection.

To make the decision maker’s problem non-trivial, we assume that (at least one) proof-

reading is ex-ante desirable. Since the expected penalty from approval of the risky project

without proofreading is calculated as Dλ, we assume that the total cost after one proofread-

ing step is smaller than Dλ:

Assumption 2. Proofreading is desirable:

Dλ> Dλ(1− p)+ c ⇔ pλ> cD

.

6

2.2 Optimal Strategy

We analyze the single decision maker’s optimal proofreading strategy. At time t, she

observed the number of detected errors by t−1, and decides whether to continue or stop the

proofreading process.

After t steps of the proofreading process, the expected penalty from implementing the

policy is:

−DE[remaining errors|history]=−DE

[M−

t∑j=1

X j

]=−Dλ(1− p)t,

where the second equality comes from Lemma 1. The total expected cost, the sum of the

expected penalty and proofreading costs of t steps, is calculated as

C(t)= Dλ(1− p)t + tc.

The decision maker’s key incentives are illustrated in Figure 1. The blue triangles denote

the benefit of approving the risky project in period t after t error-finding trials. The expected

penalty is strictly decreasing in t because the decision maker becomes more confident with

the risky policy as a longer proofreading process is conducted.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

Period (t)

Pena

lty

and

Cos

t Total cost of the projectPenalty of the project

Figure 1: A numerical example of the costs: D = 1, λ= 5, p = 12 , and c = 0.1.

The black dots represent the total expected cost of approving the risky project after

t steps of proofreading. It initially decreases in t by Assumption 2 because proofreading

decreases the number of remaining errors and the probability of having errors, and this

7

decrease dominates the constant marginal cost of additional proofreading. However, after

a certain period (t = 5 in the figure), the total expected cost increases because the marginal

benefit of having less errors is dominated by the marginal cost of additional proofreading.

We denote t∗ as the period minimizing the total expected cost.

Although it is obvious in the single decision-making problem, it will be useful in the con-

text of the collective decisions to note that the decision maker has an incentive to continue

the proofreading process during the periods in interval [0, t∗] rather than stop and approve

the risky project immediately because

Dλ(1− p)t ≥ Dλ(1− p)t∗ + (t∗− t)c.

This property will guarantee that sincere voting for the proofreading steps is a weakly dom-

inant strategy.

The optimal stopping time is determined by the cost-penalty ratio, cD . To see this point,

note that the marginal decrease of the expected number of errors must be the same as the

cost-penalty ratio at the optimal step t∗ if it were to be on R+:

λ(1− p)t∗ ln(

11− p

)= c

D.

Note that the interior solution of the above equation exists by Assumption 2. As the cost-

penalty ratio decreases, the optimal stopping time t∗ increases. Therefore, it follows that

the cost-penalty ratio is a key characteristic that represents the decision maker’s preference

over the stopping times.

3 Collective Proofreading

We extend the previous single decision maker’s problem to a collective deliberation prob-

lem. There is a decision committee consisting of n members, where n is assumed to be odd.

Each member is indexed by i ∈ N = {1, . . . ,n}. We consider the same probabilistic proofread-

ing environment. Specifically, members have the same prior distribution over the number

of errors, and the committee members jointly find errors. That is, common values of λ and

p apply to all members.

Each member’s penalty per remaining error and the proofreading cost are heteroge-

neous. This heterogeneity may be due to the members’ different positions in the agency

or political interests. For example, politicians and policymakers may have small proofread-

8

ing costs because they are less involved with the actual proofreading process compared to

engineers and researchers who actually spend their time and effort to investigate the policy.

The penalty per remaining error would also be heterogeneous: The investigator of the envi-

ronmental policy would be less affected by the remaining misconduct of the new manufac-

turing process, while the engineers who made the Space Shuttle may be critically suffered

from the remaining errors. We denote by {(D i, ci)}ni=1 the heterogeneous committee mem-

bers. Without loss of generality, we assume that D j+1 ≥ D j > 0 for j = 1, . . . ,n−1. We denote

by~t∗ = (t∗i )ni=1 the vector of each member’s optimal stopping time if they were to behave as

a single decision maker in the same proofreading environment, and we denote by m(~t∗) the

median of the optimal stopping times. We assume a nontrivial amount of heterogeneity so

that t∗i 6= t∗j for some i 6= j.Ci(t) represents member i’s expected total cost from t steps of proofreading. The ag-

gregate cost is defined as the sum of members’ costs. We assume that the socially optimal

proofreading steps coincides with at least one member’s cost minimizing proofreading steps:

Assumption 3. There exists a representative member: For some r ∈ N,

Cr(t)= 1n

n∑i=1

Ci(t).

Let t∗r = argmint∈N0 Cr(t) be the optimal stopping time of member r.3

Assumption 3 means that the aggregated cost is minimized if the committee delegates

the decision to member r.4 We call member r the representative member. All the intuitions

and results do not rely on the index (identity) of the representative member.

We consider a qualified majority rule for committee decisions. Specifically, under a q-

rule, the committee continues the proofreading process if at least q members of the commit-

tee want to do so. We call a n+12 -rule as a simple majority rule.

For any voting strategies of the other committee members, sincere voting is a weakly

dominant strategy. To understand this, observe that voting for continuation until member

i’s optimal stopping time t∗i is at least as profitable as voting against continuation before t∗i .

Voting for continuation after t∗i does not help her to attain the smallest expected cost either.

The reasoning behind this observation is the same as why a truthful bidding strategy is3Due to Lemma 1, without loss of generality, the optimal stopping time in the assumption is chosen over

the set of non-negative integers instead of the set of stopping times.4To guarantee existence of such a representative member, the assumption of no future discounting is cru-

cial. Specifically, as shown in Jackson and Yariv (2017), it is impossible to have such a representative memberif members are heterogeneous in their discount factors.

9

weakly dominant in a second-price auction. Therefore, behaving truthfully as if a committee

member were a single-decision maker is weakly dominant.

Although we do not restrict our attention to a smaller set of strategies, such a weakly

dominant strategy is symmetric, monotone, and stationary. A member’s strategy is indepen-

dent of her identity. The strategy is monotone in the sense that there is a threshold period tsuch that a member votes for the continuation of the proofreading process in or before period

t and votes against after period t. A member’s strategy does not depend on the history of

previous voting outcomes. Therefore it is straightforward to characterize a subgame-perfect

equilibrium of weakly dominant strategies.

Proposition 1. In the subgame-perfect equilibrium of weakly dominant strategies, memberi votes for continuation if and only if t ≤ t∗i .

Unless otherwise stated, we call this subgame-perfect equilibrium the equilibrium through-

out the paper. In the next section, assuming that voters act on the equilibrium, we find an

optimal voting rule.

4 Optimal Voting Rule

In this section, we analyze the equilibrium outcomes under different q-rules. We identify

the condition in which the q-rule is socially optimal.

Proposition 2. Suppose q = #{i|t∗i ≥ t∗r }. The q-rule produces the socially-optimal outcomein the equilibrium.

Proposition 2 states that the socially-optimal voting rule should make the representative

member pivotal. In other words, in any situations where the representative member is not

pivotal, collective decisions yield socially inefficient outcomes.

The key intuition for the inefficiency is captured by the fact that member i’s optimal

stopping time crucially depends on ciD i

, neither D i nor ci per se. Even if we impose a mono-

tone rank of ci, along with D i+1 ≥ D i for i = 1, . . . ,n−1, the rank of ciD i

is not sufficient to

know the optimal voting rule.

The following two examples illustrate situations in which the committee continues the

proofreading steps too much and too little under a simple majority rule, respectively.

The example in Table 1 considers a situation where the costs are negatively aligned

with the penalties (that is, ci ≥ ci+1). This cost-penalty relationship could be observed in

10

Member D i ci t∗i Ci(2) Ci(3) Ci(4) Ci(5) Ci(6) Ci(7)

1 1.0 0.60 3 2.45 2.43 2.71 3.16 3.68 4.242 1.1 0.30 4 1.98 1.59 1.54 1.67 1.89 2.143 1.2 0.15 5 1.80 1.20 0.98 0.94 0.99 1.104 1.7 0.10 6 2.33 1.36 0.93 0.77 0.73 0.775 2.0 0.05 7 2.60 1.40 0.83 0.56 0.46 0.43∑

11.15 7.98 6.99 7.09 7.75 8.67

Table 1: Inefficiency: Over-proofreading. λ= 5, p = 0.5

the committee in which senior members or authoritative members take responsibilities of

remaining errors while junior members exert effort to proofread. In this example, the order

of ciD i

is monotone, so is the optimal stopping time for each member. Although the preferences

of each member’s optimal stopping time are well ordered, and member 3 is seemingly a

median member in every aspect, the simple majority rule involves over-proofreading.

Member D i ci t∗i Ci(2) Ci(3) Ci(4) Ci(5) Ci(6) Ci(7)

1 1.0 0.05 6 1.35 0.78 0.51 0.41 0.38 0.392 1.1 0.10 5 1.58 0.99 0.74 0.67 0.69 0.743 1.2 0.20 4 1.90 1.35 1.18 1.19 1.29 1.454 1.7 0.30 4 2.73 1.96 1.73 1.77 1.93 2.175 2.0 0.40 4 3.30 2.45 2.23 2.31 2.56 2.88∑

10.85 7.53 6.39 6.34 6.85 7.62

Table 2: Inefficiency: Under-proofreading. λ= 5, p = 0.5

The example in Table 2 considers a situation where the costs are positively aligned with

the penalties (that is, ci ≤ ci+1). This cost-penalty relationship could be observed in the

committee in which some members are more involved in the policy than other members. In

this example, the simple majority rule involves under-proofreading.

Two remarks are worth mentioning. First, the positive (resp. negative) relationship

between D i and ci does not necessarily imply under-proofreading (resp. over-proofreading).

The opposite cases are also possible. Second, the optimal voting rule may require a smaller

number of votes than the simple majority to continue the proofreading process.

One remaining question would be under what conditions the simple majority rule is

socially optimal.

11

Proposition 3. The super-(sub-)majority rule is socially optimal if and only if

cm

Dm< (>)

∑ni=1 ci∑ni=1 D i

,

where the subscript m denotes the committee member whose cD is the median.

Proposition 3 also implies that the simple majority rule is socially optimal if and only ifcmDm

=∑

ci∑D i

. For example, if the cost parameters and the damage parameters are symmetri-

cally distributed, then cm and Dm coincide with the average of ci and D i, respectively, and

the simple majority rule is socially optimal. Proposition 3 indirectly illustrates how fragile

the foundation of the simple majority rule is in the context of collective proofreading deci-

sions. It is well known that the asymmetric intensities among voters may make the simple

majority rule socially undesirable.5 It is even harder to achieve efficiency in the context of

collective proofreading because we need symmetry in both dimensions.

Similar to the fact that a simple majority rule cannot be optimal for most of the cases,

any q-rule cannot be a panacea for all cases. A naturally followed question is whether

the committee could endogenously choose the optimal voting rule. In the ex-ante stage,

where every member knows the joint distribution of the damages and the costs but does not

know their realized values, the committee unequivocally prefers the optimal voting rule the

most, as it renders them the highest expected payoff. Therefore, for any voting rules and

protocols for mapping individuals’ preference orders to the committee’s preference order,

the optimal voting rule will be selected. However, in the interim stage, where committee

members privately learn their own values (D i and ci), their preferred stopping times, hence

their preferred voting rules, vary. In this case, the primitive voting rule to determine the

voting rule (Barbera and Jackson, 2004) may make a deviation from the optimal voting rule.

Thus, a normative suggestion we can draw from this section is that the voting rule should

either be made at the ex-ante stage or be exogenously determined by an unbiased executor.

5 Binomial Model

In the previous sections, thanks to the memoryless property of Poisson distributions,

each member’s preferred proofreading level was constant regardless of the number of the

previously detected errors. Since the support of the Poisson distribution is unbounded, the

model we considered in the previous sections, although it is clearer to show the inefficiency of5For the reviews of this line of research, see Posner and Weyl (2017).

12

the collective proofreading decisions, may be different from more realistic situations where

a committee collectively proofread finite issues. In this case, the number of previously de-

tected (and fixed) errors do affect further proofreading decisions. In this section, we assume

that the number of errors is drawn from a Binomial distribution.6

There is a single risky project containing M ∼ Binomial(N,π) errors, where N is the

number of issues that affect the overall return of the project, and π > 0 is the probability

that each issue contains either one error or none.

Let Yt = E[Mt|X1, . . . , X t]D+ct denote the total expected cost after t steps of proofreading,

where X j is the number of errors detected in period 1 ≤ j ≤ t, and Mt = M−∑tj=1 X j is the

number of remaining errors.

We assume the independence of the error-detecting probability:

Assumption 4. In each proofreading step, each error is detected with probability p > 0, andthe detection is independent of the detection history and detection of other errors:

Xt+1|M, X1, . . . , X t ∼Binomial(Mt, p)

Then, by Lemma 4.1 in Ferguson and Hardwick (1989), we obtain the following:

Lemma 2.

Mt|X1, . . . , X t ∼ Binomial(N −t∑

j=1X j,πt),

where the updated error probability πt is calculated as

πt = π(1− p)t

(1−π)+π(1− p)t .

There are several notable properties driven from Lemma 2. First, the updated error

probability is independent of the previously detected errors. Second, the error probability

πt is monotone decreasing in t.7 Third, the expected number of remaining errors after t6A model with this assumption nests a situation where a committee collectively decides to draw additional

signal to learn the binary state of the world, as it could be understood as an error (bad state) drawn from aBernoulli distribution.

7It directly follows from the fact that

πt = π(1− p)t

(1−π)+π(1− p)t =π

1−π(1−p)t +π

.

13

proofreading steps is history dependent:

E[Mt+1|X1, . . . , X t]= E[Mt −Xt+1|X1, . . . , X t]

= E[Mt|X1, . . . , X t]−E[Xt+1|X1, . . . , X t]

=(N −

t∑j=1

X j

)πt −E[E[Xt+1|M, X1, . . . , X t]|X1, . . . , X t]

=(N −

t∑j=1

X j

)πt −E[Mt p|X1, . . . , X t]

= (1− p)

(N −

t∑j=1

X j

)πt,

which depends on both (1) the updated error probability πt and (2) the sum of detected

errors up to period t. If other things are equal, the expected number of remaining errors is

decreasing in period t and the number of detected errors in previous periods. Consequently,

the expected cost in period t+1 is

E[Yt+1|X1, . . . , X t]= E[Mt+1|X1, . . . , X t]D+ (t+1)c

= (1− p)

(N −

t∑j=1

X j

)πtD+ (t+1)c.

The expected cost is now history dependent, so is the optimal strategy.

One assured aspect is that the optimal strategy is still monotone. In fact, it is well-

known in the optimal stopping literature that an optimal stopping strategy has a myopic

form if a stopping problem is monotone.8 The monotonicity requires that the sets

At = {Yt < E[Yt+1|X1, . . . , X t]}

are monotone increasing as At ⊆ At+1 almost surely for any t. In words, the condition

At ⊆ At+1 means that if immediate stop at time t is optimal in period t, then it is also optimal

to stop at all the following future periods, no matter how the future errors are detected.

In the current setting, the condition for monotonicity is satisfied. To see why, observe

8It is also called the one-stage look-ahead rule. See, for example, Bruss (2000).

14

that

Yt < E[Yt+1|X1, . . . , X t] ⇐⇒ E[Mt|X1, . . . , X t]pD < c

⇐⇒(N −

t∑j=1

X j

)πt pD < c

⇐⇒(N −

t∑j=1

X j

)π

1−π(1−p)t +π

pD < c.

The first term on the left-hand side of the last inequality, N −∑tj=1 X j, is the difference

between the total number of issues and the number of detected errors. This term is non-

increasing for sure. The second term, π1−π

(1−p)t+π , is the updated belief of having an error in

each remaining issue. This term is strictly decreasing and independent of the history of

detected errors. Therefore, the whole expression on the left-hand side is decreasing.

The optimal stopping time t∗ is the smallest integer t such that(N −

t∑j=1

X j

)πp

1−π(1−p)t +π

≤ cD

,

and the cost-penalty ratio, c/D is still a key characteristic. From this, we can characterize

the optimal stopping threshold as a pair of the number of detected errors and the decision

period. After rearranging the inequality, we have

t∑j=1

X j ≥ κ0 − κ1

(1− p)t ,

where κ0 = N − cD p and κ1 = c(1−π)

Dπp . This stopping rule implies that proofreading should be

stopped either when the number of detected errors is sufficiently large given the proofread-

ing steps, or when they proofread for a sufficiently long time given the number of detected

errors.

For a clearer illustration, we provide a numerical example that shows how a decision

maker’s optimal strategy depends on the detection history and steps when (N, c,D,π, p) =(12,1,8,0.5,0.5). In Figure 2, the y-axis represents the number of detected errors up to

period t,∑t

j=1 X j, and the x-axis represents the decision period t. With the set of parameters

15

specified above, we have the following optimal stopping rule:

t∑j=1

X j ≥ 9.75−2t−2.

period t

#Detected errors

STOP

STOP

continue

stop

Figure 2: Illustration of the optimal stopping rule

The blue dots in Figure 2 represents the region in which the decision maker stops proof-

reading. Observe that the monotonicity is satisfied, that is, if (x, y) is contained in the blue

region, so is (x, z) whenever z ≥ y. This monotonicity implies that Yt < E[Yt+1|X1, . . . , X t] is

also satisfied. For this feature, the boundary between the blue dots and the black dots is

decreasing in period t.Each red line represents a sample path. Each path moves to the north-east direction in

the lattice plane. The solid red line describes the scenario when X1 = 1, X2 = 2, X3 = 0, X4 =3, and the decision maker stops at period 4. The dashed red line describes the scenario when

X1 = 0, X2 = 1, X3 = 0, X4 = 0, X5 = 1, and the decision maker stops at period 5.

If t and∑t

j=1 X j were to be on R+, then we can delineate the boundary to continue proof-

reading on the plane of (t,∑t

j=1 X j). Unless otherwise stated, we call this boundary as the

optimal stopping rule. Although the optimal stopping decision for proofreading becomes

history dependent, the same results as the original model are retained. Two key features of

the optimal stopping rules are (1) that at any sample path (or history), the decision maker’s

optimal stopping rule is aligned in his/her cost-disadvantage ratio, and (2) that each hetero-

16

geneous committee’s rule does not cross each other. Committee member i with a high ciD i

will

always have a weakly lower boundary in terms of the sum of the detected errors. Since this

stopping problem is monotone, it is natural to restrict our attention to the class of mono-

tone strategies. As long as everyone uses monotone strategies, it is a weakly undominated

strategy to follow his/her own optimal stopping rule sincerely.

6 Concluding Remarks

Many serious policies and projects are implemented after rigorous proofreading steps.

When a committee with heterogeneous members collectively makes the decisions about the

proofreading process, the simple majority rule is likely to result in an inferior outcome.

The socially optimal voting rule, which requires the committee to continue the proofreading

process up to the socially optimal proofreading level can be constructed only after accounting

for the members’ heterogeneity. This construction is a nontrivial task: Although we are

completely informed about each committee member’s heterogeneity, there is no clear rule

about which voting rule should be adopted. A simple majority rule could have a bias not

only toward under-proofreading but also toward over-proofreading. The optimal voting rule

should either be made at the ex-ante stage or be exogenously determined by an unbiased

executor.

Two possible extensions are worth mentioning. First, one may consider a model in which

some members prefer taking the safe (status quo) policy rather than choosing the risky pol-

icy after they agree to stop proofreading. That is, a committee follows two-stage voting

procedures: one vote for the proofreading stringency, and the other one for approval of a

risky policy over a status quo. In this case, the representative member or the decisive

voter in the proofreading-decision stage may have to form a coalition with other members to

implement the socially-optimal policy. For this analysis, one needs to model a dynamic bar-

gaining process in line with the current proofreading process. Second, a simplified version

of the Binomial model can be a theoretical benchmark for laboratory experiments. Although

the theory has many interesting predictions, still many empirical questions remain unan-

swered: Do subjects optimally vote to stop proofreading when the expected benefit exceeds

the cost they can afford? Do they rationally respond to the changes in the distribution of the

errors of the risky project and the degree of the heterogeneity of the committee? How much

is the actual welfare loss when we stick to the simple majority voting rules that are not

17

maximizing welfare?9 We may answer all the questions by conducting controlled laboratory

experiments.

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19

A Proofs

Proof of Proposition 1

Proof. We first show that the strategy of voting for continuation if and only if t ≤ t∗i returns

a payoff at least as high as any other strategies. Let t−i be a fixed strategy profile of the

members except member i. There are two cases to consider: (1) the proofreading never

stops regardless of member i’s decision, and (2) the proofreading may stop at some period tdepending on member i’s decision in period t. Thus, it suffices to consider the second case.

Note that the members are assumed to use monotone strategies. Let t1 be the first time in

which member i’s decision matters. If t∗i ≤ t1, then the desired strategy returns the highest

payoff as it terminates the proofreading process at time t1. If t1 < t∗i , then it is better for

member i to continue the process until t∗i . Hence, the desired strategy never returns a

strictly smaller payoff than any other strategies.

We now show that there exists at least one strategy profile of other players such that the

desired strategy returns a strictly higher payoff with respect to the given strategy profile.

Let t−i be the strategy of the members except member i in which there are q−1 number of

members who vote for stop whenever t ≥ t∗i , and there are n− q number of members who

always vote for continuation. Then, it follows that member i’s decision is pivotal and playing

the desired voting strategy returns a strictly higher payoff than any other strategies.

Proof of Proposition 2

Proof. Let q = #{i|t∗i ≥ t∗r }. To show that the q-rule produces the socially-optimal outcome

in the equilibrium, it suffices to show that the proofreading process stops at time t∗r . By

Proposition 1, vote i votes for continuation whenever t ≤ t∗i . For this, for any t ≤ t∗r , there

exist at least q members who vote for continuation. Thus, the proofreading stops at exactly

period t∗r , and so the proposition is proven.

Proof of Proposition 3

Proof. Since∑n

i=1 Ci(t)= (∑n

i=1 D i)λ(1−p)t+t(∑n

i=1 ci), t∗r , the minimizing argument of∑

Ci(t),satisfies the following:

λ(1− p)t∗r ln(

11− p

)=

∑ni=1 ci∑ni=1 D i

.

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It is straightforward to check if cmDm

=∑n

i=1 ci∑ni=1 Pi

, then t∗m coincides with t∗ because

λ(1− p)t∗ ln(

11− p

)=

∑ni=1 ci∑ni=1 Pi

= cm

Pm=λ(1− p)t∗m ln

(1

1− p

).

If cmDm

<∑n

i=1 ci∑ni=1 Pi

, then t∗m > t∗. That is, the median voter prefers to proofread more than the so-

cially optimal proofreading level, and a simple majority has a bias toward over-experimentation.

The case with cmDm

>∑n

i=1 ci∑ni=1 Pi

is analogous.

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